Both above parallel samplers can be turned into parallel approximate counting algorithms, via the non-adaptive simulated annealing.
For the Ising model, using the a Chebyshev's cooling schedule in [1]} and [2]}, we can routinely turn the Ising sampler in cor:Ising to an \({O}(\log n\cdot \log \Delta )\) -depth \(\tilde{O}(mn^2/\epsilon ^2)\) -processor algorithm for \(\epsilon \) -approximation of the Ising partition function, under the same uniqueness condition.
And by the non-adaptive annealing provided in [3]}, we also have an \(\mathrm {polylog}(n)\) -depth \(\mathrm {poly}(d, n)\) -processor for approximately counting the number of satisfying solutions of any \((k,d)\) -CNF satisfying (REF ).
